Question: The lifespans of snakes in a particular zoo are normally distributed. The average snake lives $22$ years; the standard deviation is $4.1$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a snake living between $30.2$ and $34.3$ years.
Answer: $22$ $17.9$ $26.1$ $13.8$ $30.2$ $9.7$ $34.3$ $99.7\%$ $95\%$ $2.35\%$ $2.35\%$ We know the lifespans are normally distributed with an average lifespan of $22$ years. We know the standard deviation is $4.1$ years, so one standard deviation below the mean is $17.9$ years and one standard deviation above the mean is $26.1$ years. Two standard deviations below the mean is $13.8$ years and two standard deviations above the mean is $30.2$ years. Three standard deviations below the mean is $9.7$ years and three standard deviations above the mean is $34.3$ years. We are interested in the probability of a snake living between $30.2$ and $34.3$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the snakes will have lifespans within 3 standard deviations of the average lifespan. It also tells us that $95\%$ of the snakes will have lifespans within 2 standard deviations of the mean. That leaves $99.7\% - 95\% = 4.7\%$ of snakes between 2 and 3 standard deviations of the mean, or $2.35\%$ on either side of the distribution. The probability of a particular snake living between $30.2$ and $34.3$ years is $\color{orange}{2.35\%}$.